When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their exact definition or motivation. Could someone explain its motivation and definition to me? A source would also be helpful if it explains the motivation.
The sequence is [A105163]: http://oeis.org/A105163
A rather comprehensive collection of information on floretions, specifically in the context of Oeis, is Sequences related to floretions.
In essence, most of the "floretion" sequences come from an iterated function that begins with some initial value and produces one integer at each step of the iteration. The floretion algorithms operate on a 16-element quantity that can be thought of a a $4\times 4$ matrix. Such a 16-element quantity is called a floretion. The name is deliberately similar to quaternion, octonion and sedenion because the operations performed with them are similar. (But why doesn't the floretion have its own Wikipedia page?)
If you want to "hear" a floretion, here you go.
The article Structure of the Floretion Group by Richard J. Mathar defines the Floretion Group as the group of order $32$ and GAP catalogue number $49$.
The OEIS now has a much clearer definition of the general case here floretions: A308496.
Basically, floretion base vectors are numbers with digits 1,2,4,7 when written in base 8 with multiplication given via bitwise XNOR and AND operations.
****** previous answer ******
The floretion website has remained totally unchanged for the last six years- this was not due to lack of interest, but more because of my complete change of professions. As of today, I've created a new dokuwiki extending floretions from 2nd (the way they were originally defined) to 4th order, meaning there are now 256 "base vectors" instead of 16. This is all just in the infant stages and can still only be serviced in my free time.
floretion dokuwiki (Wayback Machine)
Floretions themselves are defined in several places (click on the 2nd order tab in the wiki), so I will not go into detail again here. In summary, if Q = {+-i, +-j, +-k, +-e} is the quaternion group, a "base" floretion (of which there are 16) can be written ij, ik, ei, ej, ek, ie, je, ke, ... Examples:
ij * ik = -ei
ij * ee = ij
Upon extending to 4th order floretions, we have 256 base "super floretions" which can be written
eeIJ, eiIJ, jkEJ, ..., where eeEE is the unit. Here, the first two letters are always lowercase, next two uppercase. Of course, this notation is not strictly necessary, but since the beginning I've found it helps to read from an optical standpoint.
As for the group itself, where 2nd order floretions are found to be isomorph to the factor space F = Q x Q\{(1,1), (-1,-1)}, 4th order floretions are isomorph to F x F\{(ee,ee),(-ee,-ee)} where ee is the unit. This group has 512 elements in total. Note in general when a "4th order floretion" is mentioned, we are not actually referring to the group itself, but to the 4th order floretion algebra over the reals, i.e. some vector
X = a*eeEE + b*eiEE + c*ejEE + ... with real coefficients a, b, c
Example multiplication of base vectors:
ijEK * jeKJ = <(ij*je) | (ek*kj)> = < kj | -ki > = -kjKI where ij, je, ek, kj are floretions. The complete multiplication table is given at the above link.
My main motivation is currently to find an equation similar to "Floret's Equation" for 2nd order floretions and to see what form "visual media" takes using both old and new algorithms on 4th order floretions. Currently, I've only included some files for multiplication in the project folder (written in R). However, I have begun to calculate basic algorithms and should be able to release those files very soon (I've also figured out how to "swap" elements but this needs some testing). I would be much obliged to any code reviewers / contributors.
Cheers, Creigh
